We give direct constructions of pseudorandom function (PRF) families based on conjectured hard lattice problems and learning problems. Our constructions are asymptotically efficient and highly parallelizable in a practical sense, i.e., they can be computed by simple, relatively small low-depth arithmetic or boolean circuits (e.g., in NC 1 or even TC 0). In addition, they are the first low-depth PRFs that have no known attack by efficient quantum algorithms. Central to our results is a new “derandomization ” technique for the learning with errors (LWE) problem which, in effect, generates the error terms deterministically. 1 Introduction and Main Results The past few years have seen significant progress in constructing public-key, identity-based, and homomorphic cryptographic schemes using lattices, e.g., [Reg05, PW08, GPV08, Gen09, CHKP10, ABB10a] and many more. Part of their appeal stems from provable worst-case hardness guarantees (starting with the seminal work of Ajtai [Ajt96]), good asymptotic efficiency and parallelism, and apparent resistance to quantum

Abstract. The compressed sensing problem for redundant dictionaries aims to use a small number of linear measurements to represent signals that are sparse with respect to a general dictionary. Under an appropriate restricted isometry property for a dictionary, reconstruction methods based on ℓ q minimization are known to provide an effective signal recovery tool in this setting. This note explores conditions under which ℓ q minimization is robust to measurement noise, and stable with respect to perturbations of the sensing matrix A and the dictionary D. We propose a new condition, the D null space property, which guarantees that ℓ q minimization produces solutions that are robust and stable against perturbations of A and D. We also show that ℓ q minimization is jointly stable with respect to imprecise knowledge of the measurement matrix A and the dictionary D when A satisfies the restricted isometry property. 1.

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Abstract. This article offers an accessible but rigorous and essentially self-contained account of the main ideas in compressed sensing (also known as compressive sensing or compressive sampling),

Tail inequalities for sums of random matrices that depend on the intrinsic dimension

This note presents a unified analysis of the recovery of simple objects from random linear measurements. When the linear functionals are Gaussian, we show that an s-sparse vector in R n can be efficiently recovered from 2s log n measurements with high probability and a rank r, n×n matrix can be efficiently recovered from r(6n − 5r) measurements with high probability. For sparse vectors, this is within an additive factor of the best known nonasymptotic bounds. For low-rank matrices, this matches the best known bounds. We present a parallel analysis for blocksparse vectors obtaining similarly tight bounds. In the case of sparse and block-sparse signals, we additionally demonstrate that our bounds are only slightly weakened when the measurement map is a random sign matrix. Our results are based on analyzing a particular dual point which certifies optimality conditions of the respective convex programming problem. Our calculations rely only on standard large deviation inequalities and our analysis is self-contained.

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Large data sets are often modeled as being noisy samples from probability distributions µ in R D, with D large. It has been noticed that oftentimes the support M of these probability distributions seems to be well-approximated by low-dimensional sets, perhaps even by manifolds. We shall consider sets that are locally well approximated by k-dimensional planes, with k ≪ D, with k-dimensional manifolds isometrically embedded in R D being a special case. Samples from µ are furthermore corrupted by D-dimensional noise. Certain tools from multiscale geometric measure theory and harmonic analysis seem well-suited to be adapted to the study of samples from such probability distributions, in order to yield quantitative geometric information about them. In this paper we introduce and study multiscale covariance matrices, i.e. covariances corresponding to the distribution restricted to a ball of radius r, with a fixed center and varying r, and under rather general geometric assumptions we study how their empirical, noisy counterparts behave. We prove that in the range of scales where these covariance matrices are most informative, the empirical, noisy covariances are close to their expected, noiseless counterparts. In fact, this is true as soon as the number of samples in the balls where the covariance matrices are computed is linear in the intrinsic dimension of M. As an application, we present an algorithm for estimating the intrinsic dimension of M. 1

We consider derivative-free algorithms for stochastic optimization problems that use only noisy function values rather than gradients, analyzing their finite-sample convergence rates. We show that if pairs of function values are available, algorithms that use gradient estimates based on random perturbations suffer a factor of at most √ d in convergence rate over traditional stochastic gradient methods, where d is the problem dimension. We complement our algorithmic development with information-theoretic lower bounds on the minimax convergence rate of such problems, which show that our bounds are sharp with respect to all problemdependent quantities: they cannot be improved by more than constant factors. 1

The nascent field of compressed sensing is founded on the fact that high-dimensional signals with “simple structure ” can be recovered accurately from just a small number of randomized samples. Several specific kinds of structures have been explored in the literature, from sparsity and group sparsity to low-rankedness. However, two fundamental questions have been left unanswered, namely: What are the general abstract meanings of “structure ” and “simplicity”? And do there exist universal algorithms for recovering such simple structured objects from fewer samples than their ambient dimension? In this paper, we address these two questions. Using algorithmic information theory tools such as the Kolmogorov complexity, we provide a unified definition of structure and simplicity. Leveraging this new definition, we develop and analyze an abstract algorithm for signal recovery motivated by Occam’s Razor. Minimum complexity pursuit (MCP) requires just O(κ log n) randomized samples to recover a signal of complexity κ and ambient dimension n. We also discuss the performance of MCP in the presence of measurement noise and with approximately simple signals. I.